# When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?

Document Type : Research Paper

Authors

Saurashtra University, Rajkot, India

Abstract

The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. The annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article is to classify rings  $R$ such that $(\mathbb{AG}(R))^{c}$ ( that is,  the complement of $\mathbb{AG}(R)$)   is connected and admits a cut vertex.

Keywords

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